When muscles attach to bones, they usually do so by a series of tendons. Five tendons attach to the bone, the uppermost tendon pulls at 20 degree angle from the axis of the bone, and eah tendon is directed 10 degree angle from the one next to it.
If each tendon exerts a 2.75 N pull on the bone, use vector components to find the magnitude and direction of the resultant force on this bone due to all five tendons. Let the axis of the bone be the +x axis.
Draw a graphical sum to check your results from the first part.
To find the magnitude and direction of the resultant force on the bone due to all five tendons, we can break down the forces into their x and y components and then sum them up.
First, let's denote the uppermost tendon as T1, the second tendon as T2, and so on.
The x-components of the forces are:
- T1x = 2.75 N * cos(20°)
- T2x = 2.75 N * cos(10°)
- T3x = 2.75 N * cos(0°) = 2.75 N
- T4x = 2.75 N * cos(-10°)
- T5x = 2.75 N * cos(-20°)
The y-components of the forces are:
- T1y = 2.75 N * sin(20°)
- T2y = 2.75 N * sin(10°)
- T3y = 0 N
- T4y = 2.75 N * sin(-10°)
- T5y = 2.75 N * sin(-20°)
To find the resultant force, we sum up the x and y components separately:
Sum of x-components: Rx = T1x + T2x + T3x + T4x + T5x
Sum of y-components: Ry = T1y + T2y + T3y + T4y + T5y
Let's calculate the values:
Rx = (2.75 N * cos(20°)) + (2.75 N * cos(10°)) + 2.75 N + (2.75 N * cos(-10°)) + (2.75 N * cos(-20°))
Ry = (2.75 N * sin(20°)) + (2.75 N * sin(10°)) + 0 N + (2.75 N * sin(-10°)) + (2.75 N * sin(-20°))
Using a calculator, we find:
Rx ≈ 13.95 N
Ry ≈ 2.69 N
The magnitude of the resultant force R can be found using the Pythagorean theorem:
|R| = sqrt(Rx^2 + Ry^2)
|R| = sqrt((13.95 N)^2 + (2.69 N)^2)
|R| ≈ 14.15 N
The direction of the resultant force can be obtained using the inverse tangent function:
θ = atan(Ry / Rx)
θ ≈ 11.5°
Therefore, the magnitude of the resultant force on the bone due to all five tendons is approximately 14.15 N, and the direction is about 11.5° from the +x axis.
To draw a graphical sum to check the results, you can draw vectors for each tendon, making sure the angles between them match the given values (20°, 10°, 0°, -10°, -20°). Then, you can use a ruler or protractor to measure the magnitude and direction of the resulting vector. The length of the resulting vector should approximately correspond to the calculated magnitude (14.15 N), and the direction should match the calculated direction (11.5° from the +x axis).
To find the magnitude and direction of the resultant force on the bone due to all five tendons, we can break down the force of each tendon into its vector components.
First, we need to convert the given angles to vectors. We'll use the fact that the direction of each tendon is 10 degrees from the one next to it. Starting with the uppermost tendon, which is at a 20-degree angle from the axis of the bone, we can consider it as having a component in the x-direction and in the y-direction.
Given:
Angle of the first tendon (uppermost) = 20 degrees
Angle between each tendon = 10 degrees
Force exerted by each tendon = 2.75 N
Now, let's calculate the vector components of each tendon:
1. Uppermost tendon:
- x-component = 2.75 N * cos(20°)
- y-component = 2.75 N * sin(20°)
2. Second tendon:
- x-component = 2.75 N * cos(10°)
- y-component = 2.75 N * sin(10°)
3. Third tendon:
- x-component = 2.75 N * cos(0°) (Since it has a 0-degree angle with the previous tendon)
- y-component = 2.75 N * sin(0°)
4. Fourth tendon:
- x-component = 2.75 N * cos(-10°)
- y-component = 2.75 N * sin(-10°)
5. Fifth tendon:
- x-component = 2.75 N * cos(-20°)
- y-component = 2.75 N * sin(-20°)
Now, let's calculate the resultant x-component and y-component by adding up the individual components:
Resultant x-component = sum of all x-components
Resultant y-component = sum of all y-components
Finally, we can find the magnitude and direction of the resultant force using the following equations:
Magnitude of resultant force (R) = sqrt((Resultant x-component)^2 + (Resultant y-component)^2)
Direction of resultant force (θ) = tan^(-1)(Resultant y-component / Resultant x-component)
Once you calculate the values, you can draw a graphical sum by representing the magnitude and direction of the resultant force vector using appropriate scales on a coordinate system.
To solve this problem, we will break down each tendon force into its x and y components.
Let's represent the five tendons with vectors T1, T2, T3, T4, and T5 respectively. The angle between each tendon can be represented as follows:
T1 makes a 20-degree angle with the x-axis.
T2 makes a 10-degree angle with T1.
T3 makes a 10-degree angle with T2.
T4 makes a 10-degree angle with T3.
T5 makes a 10-degree angle with T4.
First, let's determine the x and y components of each tendon:
T1x = T1 * cos(20°)
T1y = T1 * sin(20°)
T2x = T2 * cos(30°) (since it forms a 20° angle with T1)
T2y = T2 * sin(30°)
T3x = T3 * cos(40°) (since it forms a 30° angle with T2)
T3y = T3 * sin(40°)
T4x = T4 * cos(50°) (since it forms a 40° angle with T3)
T4y = T4 * sin(50°)
T5x = T5 * cos(60°) (since it forms a 50° angle with T4)
T5y = T5 * sin(60°)
Next, let's calculate the total x and y components by summing up the components of all five tendons:
Total x-component = T1x + T2x + T3x + T4x + T5x
Total y-component = T1y + T2y + T3y + T4y + T5y
Now, we can find the magnitude and direction of the resultant force by using the Pythagorean theorem and trigonometry:
Magnitude of the resultant force = sqrt((Total x-component)^2 + (Total y-component)^2)
Direction of the resultant force (relative to +x axis) = arctan(Total y-component / Total x-component)
To draw a graphical sum, you can create a coordinate system with the x-axis representing the axis of the bone. Then, draw each tendon vector at the appropriate angle and length according to their magnitudes. Finally, draw the resultant vector by connecting the initial point of the first tendon vector to the final point of the last tendon vector. The length of the resultant vector can be found by scaling it based on the magnitude of the resultant force. The angle of the resultant vector can be found using the direction calculated above.